You continue casting, up to a point
Technically, you can not "continue casting a spell" but "do not spend your action". You spend an action because you're casting a spell, not vice versa. The same way we use "Attack action" and "attack roll" to simulate combat, for instance — you make these rolls because you're attacking an enemy.
That's why this comment makes sense:
It says you must spend your action each turn, but it does not say that if you do not spend an action that the spell fails
"Casting a spell" takes place in the game world. "Action" does not exist in the game world. We use game mechanics (actions) to model in-game events (spellcasting).
Let's take a specific example. The Wizard is channeling a spell. Suddenly, goblins attack. The Wizard is busy and distracted, so he is "surprised". In the game world that means he is not ready for the fight yet. So, what does he do instead? He either fumbles, or continue the channeling. This is what happens in the game world. This particular situation is quite rare and is not described in the rules, hence, requires DM's adjudication. That's why the game requires a DM, after all.